In Theorems REF  and REF  we prove global well-posedness in the defocusing case. Global well-posedness in \(L^2\) -based Sobolev spaces typically follows from conserved quantity. This does not fit well into the \(L^p\) -scale. Instead, we show global well-posedness by adapting arguments of Dodson–Soffer–Spencer [1]}; see also [2]}, [3]}.
